Composite Number Tree
Each number branches from its largest divisor
The number line has a fractal structure
The integers have a beautiful fractal structure that can be seen by sorting them in
terms of divisibility. The Number Tree is constructed using a simple rule:
every integer (except 0 and 1) branches off from its largest divisor.
consider the number 20. It has five divisors (not counting itself): 1, 2, 4, 5, and 10.
10 is the largest divisor, and so the number 20 branches from the number 10. Now consider the number 21. It
has 1, 3 and 7 as its divisors, and so it branches from 7. The number 22 branches from 11. And the number
23 (a lonely prime) branches from 1.
Notice how a fully-grown tree is bushier on the left, and a spiral
of prime numbers extends out at lower-right.
This is due to the ordering of the numbers, which is expressed in terms of angles.
The position of a given number
> 1) is based on polar coordinates:
(shown as 'distance arcs') and
(shown as 'order rays'), centered at the number 1. These value can be adjusted using the controls for 'angle spread' and 'growth exponent'.
The angular value of
is determined by the particular ordering of the numbers,
which in turn is determined by the way new numbers are inserted into the ordered array
of numbers, as the tree grows.
Every new number is inserted at an order-position adjacent to its largest divisor (one order-unit
counter-clockwise). Using this simple rule, patterns emerge as the tree grows. Branch-points
represent multiplication by primes.
The highly-composite numbers are concentrated
at the lower-left, as well as power series (such as the unbranchable 2n branch at bottom-left).